let $f_1(x,y),f_2(x,y),\dots{}$ be an infinitely countable number of polynomials in $\mathbb{R}[x,y]$. Denote by $C_n$ the curve in $\mathbb{R}^2$ given by $$ C_n = \bigl\{ (a,b) \in \mathbb{R}^2 \mid f_n(a,b) = 0 \bigr\}. $$
If $C$ is the intersection of all curves, that is,$$ C= \bigcap^{\infty}_{n=1}C_n$$ prove that there exists $m\geq1$ such that $$ C = \bigcap^{m}_{n=1}C_n, $$ this is the intersection of a finite number of curves.
I think we can use Hilberts Basis Theorem, but I can't think of how to start the argument. Could you help me, please!
Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.
Continuing the argument of comments, let us consider the chain $$(f_1)\subset (f_1,f_2)\subset\cdots$$ which is stationary, say, at $(f_1,\dots,f_m)$. That is, $f_j\in (f_1,\dots,f_m)$ and $$f_j=a_1^{(j)}f_1+\cdots+a_m^{(j)}f_m\quad (*)$$ holds for all $j=m+1,m+2,\cdots$. Now, for the intersection, $$\bigcap_{n=1}^{\infty} C_n\subset \bigcap_{n=1}^m C_n$$ holds trivially. The converse also holds since if $(x,y)$ is in RHS then by the relation $(*)$ it automatically satisfy $f_j(x,y)=0\ (j=m+1,m+2,\dots )$, hence it is in LHS, too.